The Algebra of Multitangent Functions

The Algebra of Multitangent Functions

Olivier Bouillot

D´epartement de Math´ematiques Bˆatiment425

Facult´e des Sciences d’Orsay Universit´e Paris-Sud 11

F-91405 Orsay Cedex

Abstract

Multizeta values are numbers appearing in many different contexts. Un- fortunately, there arithmetics remains mostly out of reach.

In this article, we define a functional analogue of the algebra of mul- tizetas values, namely the algebra of multitangent functions, which are 1- periodic functions defined by a process formally similar to multizeta values.

We introduce here the fundamental notions of reduction into monotan- gent functions, projection onto multitangent functions and that of trifac- torisation, giving a way of writing a multitangent function in terms on Hurwitz multizeta functions. This explains why the multitangent algebra is a functional analogue of the algebra of multizeta values. We then discuss the most important algebraic and analytic properties of these functions and their consequences on multizeta values, as well as their regularization in the divergent case.

Each property of multitangent has a pendant on the side of multizeta values. This allowed us to propose new conjectures, which have been checked up to the weight 18.

Keywords: Multizetas values, Hurwitz multizeta values, Eisenstein series, Multitangents functions, Reduction into monotangents, Quasi-symmetric functions, Mould calculus.

2010 MSC:11M32, 11M35, 11M36, 05M05, 33E20.

Email address: olivier.bouillot@math.u-psud.fr(Olivier Bouillot) URL: http://www.math.u-psud.fr/∼bouillot/(Olivier Bouillot)

Preprint submitted to Elsevier April 24, 2013

Contents

1 Introduction 3

1.1 The Riemann zeta function at positive integers . . . 3

1.2 The multizeta values . . . 4

1.3 On multitangent functions . . . 6

1.4 Eisenstein series . . . 8

1.5 Results proved in this article . . . 9

2 Definition of the multitangent functions and its first prop- erties 12 2.1 A lemma on symmetrel moulds . . . 12

2.2 Application: definition of multitangent functions . . . 18

2.3 First properties of multitangent functions . . . 19

3 Reduction into monotangent functions 20 3.1 A partial fraction expansion . . . 20

3.2 Expression of a multitangent function in terms of multizeta values and monotangent functions . . . 21

3.3 Tables of convergent multitangent functions . . . 23

3.4 Linear independence of monotangent functions . . . 23

3.5 A first approach to algebraic structure of MT GF . . . 24

3.6 Consequences . . . 25

4 Projection onto multitangent functions 26 4.1 A second approach to the algebraic structure of MT GF . . 26

4.2 A structure of MZV-module . . . 27

4.3 About the projection of a multizeta value onto multitangent functions . . . 28

4.4 About unit cleansing of multitangent functions . . . 32

5 Algebraic properties 35 5.1 Is MT GF_CV a graded algebra ? . . . 35

5.2 On a hypothetical basis of MT GF_CV,p . . . 39

5.3 The Q-linear relations between multitangent functions . . . 41

5.4 On the possibility of finding relations between multizeta val- ues from the multitangent functions . . . 43

5.5 Back to the absence of the monotangent Te¹ in the relations of reduction . . . 45

2

6 Analytic properties 50

6.1 Fourier expansion of convergent multitangent functions . . . 50

6.2 An upper bound for multitangent functions . . . 52

6.3 About the exponentially flat character . . . 55

7 Study of a symmetrel extension of multitangent functions to seq(N^∗) 57 7.1 A generic method to extend the definition of a symmetrel mould . . . 58

7.2 Trifactorization of Te^• and consequences . . . 62

7.3 Formal Hurwitz multizeta functions and formal multitangent functions . . . 64

7.4 Properties of the extension of the mould Te^• to seq(N^∗) . . . 67

7.5 Reduction into monotangent functions . . . 70

8 Some explicit calculations of multitangent functions 80 8.1 Calculation of Te^1[r](z) , for r∈N. . . 80

8.2 Calculation of Te^n[k](z), forn ∈N^∗ and k∈N . . . 82

8.3 About odd, even or null multitangent functions . . . 89

8.4 Explicit calculation of some multitangent functions . . . 90

9 Conclusion 91 A Introduction to mould notations and calculus 93 A.1 Notion of moulds . . . 93

A.2 Mould operations . . . 94

A.3 Symmetrality . . . 95

A.4 Symmetrelity . . . 96

A.5 Symmetrility . . . 97

A.6 Some examples of rules . . . 98

A.7 Some notations . . . 98 1. Introduction

1.1. The Riemann zeta function at positive integers

An interesting problem, but still unsolved and probably out of reach today, is to determine the polynomial relations overQbetween the numbers

3

ζ(2) , ζ(3) , ζ(4) , · · · , where the Riemann zeta functionζ can be defined by the convergent series

ζ(s) =

+∞

X

n=1

1 n^s in the domain <e s >1 .

Thanks to Euler, we know the classical formula for all even integerss:

ζ(s) =−(2π)^s 2

|B_s| s! .

From this, one can see that Q[ζ(2), ζ(4), ζ(6),· · ·] = Q[π²] . Now, Linde- mann’s theorem on the transcendence of π concludes the discussion for s even, as the last ring is of transcendence degree 1 .

Euler failed to give such a formula for ζ(3). Actually, the situation is quite more complicated concerning the values of the Riemann zeta function at odd integers. Essentially, nothing is known about their arithmetics. One had to wait the end of the twentieth century to see the first results:

1. In 1979, Roger Ap´ery proved that ζ(3) is an irrational number (see [1]) ;

2. In 2000, Tanguy Rivoal proved there are infinitely many numbers in the list ζ(3) ,ζ(5) , ζ(7) ,· · · which are irrational numbers (see [29]) ; 3. in 2004, Wadim Zudilin showed that there is at least one number in

the list ζ(3) , ζ(5) , · · · , ζ(11) which is irrational (see [40]) .

One conjectures that each number ζ(s) , s ≥ 2, is a transcendental number. To be more precise, the following conjecture is expected:

Conjecture 1. The numbers π , ζ(3) , ζ(5) , ζ(7) , · · · are algebraically independent over Q .

1.2. The multizeta values

The notion of multizeta value has been introduced in order to study questions related to this conjecture. Multizeta values are a multidimen- sional generalization of the values of the Riemann zeta functionζat positive integers, defined by:

Ze^s1,···,sr = X

0<nr<···<n1

1

n₁^s1· · ·n_r^sr , for all sequences ofS_b^? ={(s₁;· · · ;s_r)∈seq(N^∗) ;s₁ ≥2}.

4

Their first introduction dates back to the year 1775 when Euler studied in his famous article [21] the case of length 2. In this work, he proved numerous remarkable relations between these numbers, likeZe^2,1 =Ze³ or more generally:

∀p∈N^∗ ,

p−1

X

k=1

Ze^p+1−k,k =Ze^p+1 .

Although they sporadically appeared in the mathematics as well as in the physics literature, we can say that they were forgotten during the XIX^th century and during most of the XX^thcentury. In the last 70’s, these numbers have been reintroduced by Jean Ecalle in holomorphic dynamics. He used them as auxiliary coefficients in order to construct some geometrical and analytical objects, such as solutions of differential equations with specific dynamical properties. During the late 80’s, multizeta values appeared in many different contexts. They have been the object of an enormous renewed interest, which has then been massive and decisive. Finally, these numbers began to be studied for themselves.

Today, multizeta values arise in many different areas like in:

1. Number theory (search for relations between multizeta values, in or- der to study the hypothetical algebraic independence of values of Rie- mann’s zeta function ; arithmetical dimorphy) : see [17], [36], [39] for example.

2. Quantum groups, knot theory or mathematical physics (with the Drin- feld associator which has multizeta values as coefficients): see [5], [6], [24] ou [26].

3. Resurgence theory and analytical invariants (in many cases, these in- variants are expressed in term of series of multizetas values) : see [3]

and [4]

4. the study of Feynman diagrams : see [5], [6] or [26].

5. the study ofP¹−{0; 1;∞}(through the Grothendieck-Ihara program):

see [23], [25], [27] for example.

6. the study of the “absolute Galois group”: see [24] for example.

In regard of Conjecture 1, one of the important questions is the under- standing of the relations between multizeta numbers. There are numerous relations between these numbers, coming in particular from their represen- tation as iterated series or as iterated itegral. Let us remind what is the seond representation.

5

It is now a well-known fact that multizeta values has a representation as an iterated integral. This can be seen in the following way. If we consider the 1-differential forms

ω₀ = dt

t and ω₁ = dt 1−t , the iterated integral

Wa^{α1,···,αr} = Z

0<t1<···<tr<1

ω_α1· · ·ω_αr

is well defined when (α₁;· · · ;α_r) ∈ {0; 1}^r satisfied α₁ = 1 and α_r = 0 . This allows us to defined a symmetral mould, denoted by Wa^• .

It is easy to see that the moulds Ze^• and Wa^• are related each others by :

Ze^s1,···,sr =Wa^{1,0[sr−1],···,1,0[s1−1]} , for all sequences (s₁;· · ·;s_r)∈seq(N^∗) stisfying s₁ ≥2 .

Among others, the relation coming from the symmetrality and sym- metrelity relations are particularly important. These two types of relations allow us to express a product of two multizeta values as aQ-linear combi- nation of multizeta values in two different ways. One conjectures that these two families (up to a regularization process) spans all the other relations between these numbers (see [36] or [39]). This conjecture, out of reach to- day, would in particular show the absence of relations between multizeta values of different weights, and so the transcendence of the numbersζ(s) , s≥2 .

1.3. On multitangent functions

In this article, we will present an algebra of functions, the algebra of multitangent functions, which is in a certain sense a good analogue of the algebra of the multizeta values. Before we give the definition, let us mention two ideas which underlie the definition of multitangent functions.

First, we know that one of the essential ideas of the explicit calculation of ζ(2n) , where n ∈ N, is a symmetrization of the set of summation, that is to say a transformation which allows us to transform a sum overN into a sum over Z . Here, the transformation comes from the expansion of the cotangent function. By the same idea, we are able to compute numerous sums of the form X

m∈N^∗

ω^mr

m^r , where ω is a root of unity.

6

Consequently, it is a natural idea to try to symmetrize the summation simplex of multizeta values.

Next, some well-known ideas are interesting to stress out. One knows that working with numbers imposes a certain rigidity, while working with functions, which will be evaluated after to a particular point, gives more flexibility. One also knows that working with periodic functions gives us access to a whole panel of methods.

The simplest suggestion of a functional model of multizeta values is to consider the Hurwitz multizeta functions:

z 7−→ He^s₊^1,···,sr(z) = X

0<n1<···<nr

1

(n₁+z)^s1· · ·(n_r+z)^sr ,

for all sequences of (s₁,· · · , s_r)∈ S_b^? . The advantage of these functions is to have a very simple link with the multizetas values:

He^s₊^1,···,sr(0) =Ze^s1,···,sr , where (s1,· · · , sr)∈ S_b^? .

Unfortunately, this choice seems not to be the best one, according to the previous remarks: these functions are not periodic and the set of summation is not symmetric... So, we are led to modify the model by considering the functions:

z 7−→ Te^s1,···,sr(z) = X

−∞<n₁<···<n_r<+∞

1

(n₁+z)^s1· · ·(n_r+z)^sr , for all sequences ofS_b,e^? ={(s₁;· · ·;s_r)∈seq(N^∗) ;s₁ ≥2 and s_r ≥2}.

Obviously, these are 1-periodic functions and the set of summation is a symmetric set. Nevertheless, what is gained on one side is obviously lost on the other one: in spite of similar expressions, the link with multizeta values is not so clear. Indeed, this link does exist and is actually stronger than the one with Hurwitz multizeta functions (see§3 and §7.5.3) .

We are going to refer to these functions as “multitangent functions”.

The prefix “multi” characterizes the summation set in more than one vari- able; the suffix “tangent” comes from the link between Einsenstein series and the cotangent function. A more representative name would have been

7

“multiple cotangent functions” or “multicotangent functions”, but we pre- ferred to simplify it by forgetting the syllable “co”, which doesn’t alter its quintessence.

To the best of our knowledge, this family of functions had never been studied from the point of view of special functions, even if it is an interesting and completely natural mathematical object. There are, actually, three good reasons to study such a family of functions, in an algebraic as well as in an analytical way:

1. The multitangent functions seem to have appeared for the first time in resurgence theory and holomorphic dynamics, in a book of Jean Ecalle (see [14], vol. 2 as well as [3] or the survey [4]). Consequently, these functions have some direct applications.

2. The multitangent functions are deeply linked to multizeta values, at least because of an evidence formal similarity. In a naive approach, we can raise the same questions as for multizeta values, but this time for multitangent functions.

3. The multitangent functions are a multidimensional generalization of the Eisenstein series, which have been used by Eisenstein to develop his theory of trigonometric functions in his famous article of 1847 (see [20] or [37] for a modern approach). So, interesting facts may emerge from this generalization.

1.4. Eisenstein series

The series considered by Eisenstein are defined for all z ∈C−Z by:

ε_k(z) = X

m∈Z

1 (z+m)^k , wherek ∈N^∗ .

As Eisenstein himself said, “the fundamental properties of these simply- periodic functions reveal themselves through consideration of a single iden- tity” (see [20]):

1

p²q² = 1 (p+q)²

1 p² + 1

q²

+ 2

(p+q)³ 1

p+ 1 q

.

From this, he would obtain some identities, which are non trivial at a first sight, between these series. About the ingenuity and the virtuosity of

8

Eisenstein, Andr´e Weil compared his work with one of the most difficult works, even today, of the last period of creation of Beethoven: the Diabelli variations. It is a work of art based from the most harmless theme which may be and which, during the variations following one another, will generate a prodigious and extremely rich musical universe which is full of delicacy, but also at the same time full of pianos and compositional virtuosity. The parallel to show the beauty of the results obtained by Eisenstein is crystal clear.

In his variations, Eisenstein obtained, in particular, the following rela- tions:

ε₂²(z) = ε₄(z) + 4ζ(2)ε₂(z) . (1)

ε3(z) = ε1(z)ε2(z). (2)

3ε₄(z) = ε₂(z)²+ 2ε₁(z)ε₃(z) . (3) Eisenstein also proved that each of his series is in fact a polynomial with real coefficients inε₁ . In our study of the algebraic relations between multitangent functions, we will find another proof of the relations (1) , (2) and (3) . These are particular cases of more general relations: the relation (1) is a mix of the relations of symmetrelity and of the reduction of multitangent function into monotangent functions, while relations (2) and (3) are the archetype of relations of symmetrelity for divergent multitangent functions.

Let us mention that although Weil preferred in [37] the notation ε_k in honour of Eisenstein, from now on, we will systematically use the notation Te^scoming from multitangent functions. Also, in connection with the name

“multitangent functions”, we shall name them “monotangent functions” in order to mean the sequence is of length one.

1.5. Results proved in this article

Because of the three fundamental reasons evoked before, we have ini- tiated a complete study of multitangent functions. The first important properties (see§2 and 5) are:

Property 1. 1. The mould¹ Te^• of multitangent function is a symmetrel mould, that is, for all sequences ααα and βββ in S_b,e^? , we have

Te^ααα(z)Te^βββ(z) = X

γ∈she(ααα,βββ)

Te^γγγ(z) , where z ∈C−Z . 9

2. There are many Q-linear relations between of multitangent functions.

In one word, the first point allows us to find more than one half of all the known algebraic relations between multizeta values (the relation of symmetrelity and a few of double-shuffle relations), while the second point allows us to find exactly the others algebraic relations between multizeta values (the relation of symmetrality and the other double-shuffle relations).

We will also see that each multitangent function has a simple expres- sion in term of multizeta values and monotangent functions. We will also determine that a sort of converse is true: the algebra of multitangent func- tions is a module over the algebra of multizeta values. The first property is called the “reduction into monotangent functions” (see§3), while the second property is called “projection onto multitangent functions” (see §4).

Theorem 1. (Reduction into monotangent functions)

For all sequences s= (s1;· · · ;sr)∈seq(N^∗), there exists an explicit family (z^s_k)k∈[[ 0 ;M]] ∈ V ect_Q(Ze^σ)σ∈S_b^?

M+1

, with M = max

i∈[[ 1 ;r]]s_i, such that:

Te^s(z) =z₀^s+

max(s1;···;sr)

X

k=1

z_k^sTe^k(z) , where z ∈C−Z . Moreover, if s∈ S_b,e^? , then z₀^s =z₁^s = 0 .

From an algebraic point of view, let us define some algebras related to the first point of the property 1:

MZV_CV = Vect_Q(Ze^s)_s∈S?

b and MZV_CV,p = Vect_Q(Ze^s)s∈S? b

||s||=p

, HM ZF_CV,+ = Vect_Q(He^s₊)_s∈S?

b

and HM ZF_CV,+,p= Vect_Q(He^s₊)s∈S?

||s||=pb

,

HM ZFCV,− = Vect_Q(He^s₋)_s∈S?

e and HM ZFCV,−,p= Vect_Q(He^s₋)s∈S?

||s||=pe

, MT GF_CV = Vect_Q(Te^s)_s∈S?

b,e and MT GF_CV,p= Vect_Q(Te^s)s∈S? b

||s||=p

,

HM ZVCV,±= Vect_Q

He^s₊¹He^s₋²

s1∈S? b s2∈S? e

,

1See the appendix for a brief introduction to mould notations and calculus.

10

where p ∈ N, S_e^? = {(s₁;· · · ;s_r) ∈ seq(N^∗) ;s_r ≥ 2} and the weight of a sequences= (s1,· · · , sr)∈N^? is defined by:

||s||=s₁+· · ·+s_r . Using this notation, we can state the following:

Theorem 2. (Projection onto multitangent functions) The following assertions are equivalent:

1. For all non negative integer p , MT GF_CV,p =

p−2

M

k=0

MZVCV,p−k· Te^k . 2. MT GF_CV is a MZV_CV-module.

3. For all sequence σσσ∈ S_e^? , Ze^σσσTe² ∈ MT GFCV,||σσσ||+2 .

We will see that the duality reduction/projection is a very important process (see§5). In one sentence, we can sum up all the study by saying:

“the algebra of multitangent functions is a functional analogue of the algebra of multizeta values: each result on multizeta values has a translation in the algebra of multitangent functions, and conversely.”

We can also sum up this study by the following diagram:

MZV_CV

projection

HM ZF_+,CV

evaluation at 0

oo  _

MT GF_CV

reduction

OO

 trifactorization //HM ZF±,CV

In this diagram, which will be constructed throughout the article as an evolutive one, the trifactorization is an explicit expression of each multi- tangent function in term of Hurwitz multizeta functions. Using it, we will be able to regularize divergent multitangent functions (see §7), that is to say multitangent functions depending of a sequence s ∈ N^?1 − S_b,e^? . This explains that we allow such sequences in the Theorem 7 .

11

We will also see some analytical properties of the multitangent functions (see 6), such as their Fourier expansion or their upper bound on the half- plane, which would be useful for direct applications. Finally, we will perform some explicit calculation (see section 6) to obtain:

Property 2. Let n ∈N^∗ and k ∈N.

Let us also set E the floor function and define for (k;n) ∈ N ×N^∗ the functions t_k,n by:

∀x∈R , t_k,n(x) =

cos⁽ⁿ⁻¹⁾(x) , if k is odd.

sin⁽ⁿ⁻¹⁾(x) , if k is even.

Then, we consider the moulds sg^• , e^• and s^• , which are C-valued and defined over the alphabetΩ = {1;−1}:

sg^ε =

n

Y

k=1

ε_k, s^ε=

n

X

k=1

ε_k, e^ε =

n

X

k=1

ε_ke^(2k−1)iπn.

Then, for all z ∈C−Z , we have:

Te^n[k](z) = (−1)^{n−1+E(kn+12 )}π^kn (kn)!(2 sin(πz))ⁿ

X

ε=(ε1;···;εn)∈Ωⁿ

sg^ε(e^ε)^kntkn,n(s^επz) .

2. Definition of the multitangent functions and its first properties Let us begin with a general lemma which immediately shows, if a certain condition holds, that a mould defined as an iterated sum of holomorphic functions is a symmetrel mould valued in the algebra of holomorphic func- tions. This will give us the analytical definition of multitangent functions, but this will also be useful for dealing with the Hurwitz multizeta func- tions in the sequel. In the case of multizeta values, it gives the well-known convergence criterion.

As a consequence of this lemma, we will obtain four elementary, but fundamental, properties of multitangent functions.

2.1. A lemma on symmetrel moulds

This is a first version of this lemma, for classical sums, that is to say when the summation index varies from N to +∞, when N ∈N:

12

Lemma 1. (Definition of symmetrel moulds, version 1.)

Let U be an open set of C, (f_n)n∈N a sequence of holomorphic functions on U and N ∈N .

We assume that for all compact subsets K of U,

||fn||∞,K =

n−→+∞O

1 n

.

Then, for all sequences s∈seq(C)− {∅}, of length r, satisfying







<e(s₁)>1, ...

<e(s₁+· · ·+s_r)> r,

(4)

we have:

1. The function F e^s_N : U −→ C

z 7−→ X

N <nr<···<n1<+∞

(f_n1(z))^s1· · ·(f_nr(z))^sr is well defined on U.

2. F e^s_N is holomorphic on U and for all z ∈ U: (F e^s_N)⁰(z) = X

N <nr<···<n1<+∞

r

Y

i=1

(f_ni)^si

!0

.

Moreover, F e^•_N a symmetrel mould defined on the set of sequences s ∈ seq(C) satisfying (4), valued in H(U) , if we set F e^∅_N = 1 .

All the interest of this lemma is to give in one result an absolute conver- gence criterion for iterated sum as well as to give the symmetrel character.

So, from now on, each time we will consider a mould which satisfies the hypothesis of this lemma and its second version, we will just say it will be a symmetrel mould without further explanation.

In the following proof, we will just indicate the reason of the conditions imposed to obtain absolute convergence of the series and the holomorphy ofF e^s_N. Nevertheless, we will prove in detail the symmetrelity ofF e^s_N even if it is also elementary and a direct consequence of a calculation made by Michael Hoffman (see [22], page 485) .

13

Proof. Points 1 and 2 can be proved simultaneously because the series which defines F e^s_N is normally convergent on every compact subset of U . Thus, the classical theorem of Weierstrass for limit of sequences of holo- morphic functions concludes the proposition. Actually, if K is a compact subset ofU , there exists MK >0 such that for all n∈N:

||f_n||∞,K ≤ M_K n+ 1 .

Besides, for z ∈ K, we can write fn(z) = rn(z)e^iθn(z) with rn(z) ≥ 0 and θ_n(z)∈]−π;π] . Thus: |f_n(z)ⁱ|=e^−θn(z)∈[0;e^π] .

In particular, we obtain: |f_n(z)^s| ≤ M_K^{<e s}e^{π=m s}

(n+ 1)^{<e s} . Therefore, there exists a constantC >0 satisfying:

X

N <nr<···<n1

||f_n^s1₁· · ·f_n^sr_r||∞,K ≤ X

N <nr<···<n1

C

(n₁+ 1)^{<e s1}· · ·(n_r+ 1)^{<e sr}

≤ CZe^{<e s1,···,<e sr} <+∞ .

LetN ∈N . We will show the symmetrelity of F e^•_N(z) by an induction process. To be precise, we will show the equalityF e^s_N¹(z)F e^s_N²(z) = X

γ∈she(s¹;s²)

F e^γ_N(z) , with sequencess¹ ands² of seq(C) satisfying (4) . The induction is over the

integer l(s¹) +l(s²) .

Before starting², let us remind that, if s ∈ seq(C) satisfy (4), then, by definition of F e^•_N, we have:

F e^s_N = X

p>N

(f_p)^skF e^s_p^<r.

Anchor step: Let (u;v)∈(seq(C))² satisfying (4) and l(u) =l(v) = 1.

14

Writingu= (u) andv= (v), we successively have, for N ∈N: F e^u_NF e^v_N = X

p>N

(f_p)^u

! X

q>N

(f_q)^v

!

= X

p>q>N

(fp)^u(fq)^v+ X

p=q>N

(fp)^u(fq)^v+ X

q>p>N

(fp)^u(fq)^v

= X

q>N

(f_q)^vF e^u_q(z) +F e^u+v_N (z) +X

p>N

(f_p)^uF e^v_p(z)

= F e^u,v_N +F e^u+v_N +F e^v,u_N = X

w∈she(u;v)

F e^w_N .

Thus, the property is initialised.

Induction step: Let us suppose that the result is proved for all sequencesu and vof seq(C) satisfying (4) and such that l(u) +l(v)≥2 .

In the same way as for length 1 and by the use of the induction hypothesis, if u and v are of length k and l respectively, we successively have:

2Let us remind that if s = (s₁,· · ·, s_r), the notation s^≤k refers to the sequence (s₁,· · ·, s_k) of the firstkterms ofs, whiles^<krefers to the empty sequence when k= 1 or the sequence of the first (k−1) terms ofsifk≥2 .

For this notation, see the annex on mould calculus.

15

F e^u_NF e^v_N = X

p>q>N

(f_p)^uk(f_q)^vlF e^u_p^≤k−1F e^v_q^≤l−1+ X

n=p=q>N

(f_n)^uk+vlF e^u_n^≤k−1F e^v_n^≤l−1

+ X

q>p>N

(fp)^uk(fq)^vlF e^u_p^≤k−1F e^v_q^≤l−1

= X

q>N

(f_q)^vlF e^v_q^≤l−1F e^u_q +X

n>N

(f_n)^uk+vlF e^u_n^≤k−1F e^v_n^≤l−1

+X

p>N

(f_p)^ukF e^u_p^≤k−1F e^v_p

= X

w∈she(u;v^≤l−1)

X

q>N

(f_q)^vlF e^w_q

!

+ X

w∈she(u^≤k−1;v^≤l−1)

X

n>N

(f_n)^uk+vlF e^w_n

!

+ X

w∈she(u^≤k−1;s)

X

p>N

(f_p)^ukF e^w_p

!

= X

w∈she(u;v^≤l−1)·vl

F e^w_N + X

w∈she(u^≤k−1;v^≤l−1)·(uk+vl)

F e^w_N + X

w∈she(u^≤k−1;v)·uk

F e^w_N

= X

w∈she(u;v)

F e^w_N .

Thus, by induction, for all sequences s¹ and s² of seq(C) satisfying (4), we have:

F e^s_N¹F e^s_N² = X

γγγ∈she(s¹;s²)

F e^γγγ_N where N ∈N .

In other words, for all z ∈ U and N ∈ N, the mould F e^•_N(z) is a symmetrel one.

We obtain, as a corollary, the second version of this lemma, but for sums over all integers:

Lemma 2. (Definition of symmetrel moulds, version 2.)

Let U be an open set of C, (f_n)n∈Z a sequence of holomorphic functions 16

on U .

We assume that for all compact subsets K of U,

||fn||∞,K =

n−→±∞O

1

|n|

.

1. Then, for all sequences s∈seq(C)− {∅}, of length r, satisfying

∀k∈[[ 1 ; r]],

<e(s1+· · ·+sk)> k,

<e(s_r+· · ·+sr−k+1)> k, (5) the function F e^s : U −→ C

z 7−→ X

−∞<nr<···<n1<+∞

fn1(z)s1

· · · fnr(z)sr

is well defined on U, holomorphic on U and satisfy:

∀z ∈ U,(F e^s)⁰(z) = X

−∞<nr<···<n1<+∞

r

Y

i=1

(f_ni(z))^si

!0

.

2. Moreover, F e^•_N a symmetrel mould defined on the set of sequences s∈seq(C) satisfying (12), valued inH(U) , if we set F e^∅_N = 1 . Proof. The lemma for definition of symmetrel moulds, version 1, has sev- eral consequences:

•The mould F e^• can be factorised:

F e^•(z) = F e^•₊(z)×Ce^•(z)×F e^•₋(z) , (6) where, for all s ∈ seq(C) satisfying (5), the functions F e^s₊ , Ce^s and F e^s₋ are defined on U by:

F e^s₊(z) = X

0<nk<···<n₁<+∞

l(s)

Y

i=1

(f_ni(z))^si .

Ce^s(z) =







1 , if l(s) = 0 . (f₀(z))^s1 , if l(s) = 1 . 0 , otherwise.

F e^s₋(z) = X

−∞<nr<···<n_k<0 l(s)

Y

i=k

(f_ni(z))^si . 17

Actually, let us set F e^s₊₀(z) = X

0≤nr<···<n1<+∞

l(s)

Y

i=k

(fni(z))^si where z ∈ U and s ∈ seq(C) satisfying (5) . In the definition of F e^•₊₀(z), we obtain by isolating the summation indexn_r when it is equal to 0:

F e^s₊₀(z) = X

0=nr<nr−1<···<n1<+∞

l(s)

Y

i=k

(fni(z))^si + X

0<nr<nr−1<···<n1<+∞

l(s)

Y

i=k

(fni(z))^si

= (f₀(z))^srF e^s₊^≤r−1(z) +F e^s₊(z) = F e^•₊(z)×Ce^•(z)s

.

In the same way, we show that F e^•(z) =F e^•₊₀(z)×F e^•₋(z), which im- plies the trifactorisation (6) .

• Since s ∈ seq(C) satisfies (5), s and ^←s satisfy (4) . The lemma for definition of symmetrel moulds, version 1, shows us that the functionsF e^s₊^≤k andF e^s₋^≥k are well defined and holomorphic onU and that their derivatives can be calculated by a term by term process.

Thus,F e^s is well defined and holomorphic onU, with a derivative which is the summation of the summand derivatives.

•Moreover, according to the first version of this lemma, F e^•₊ and F e^•₋ are symmetrel moulds, as well as Ce^• . Since the mould product of symmetrel moulds defines a symmetrel mould, we deduce that F e^• is a symmetrel mould for allz ∈ U .

2.2. Application: definition of multitangent functions Let us considerU =C−Z and for n∈Z, the functions

f_n: U −→ C

z 7−→ 1

n+z . It is clear that, for all compact subsetsK of C−Z,

||f_n||_∞,K =

n−→±∞O

1

|n|

. 18

The lemma for definition of symmetrel moulds, version 2, allows us to define a symmetrel mould, denotedTe^•, defined by:

Te^s: C−Z −→ C

z 7−→ X

−∞<nr<···<n₁<+∞

1

(n₁+z)^s1· · ·(n_r+z)^sr.

This mould, which will be called the mould of multitangent functions, is defined, a priori, for all sequences

s∈ S_b,e^? =

s∈seq(N^∗);s₁ ≥2 and s_l(s) ≥2

and is valued in the algebra of holomorphic functions defined on C−Z. 2.3. First properties of multitangent functions

Here are the first elementary properties satisfied by the multitangent functions. These are consequences of Lemma 2 or of a simple change of variables in the summations (third point):

Property 3. 1. The function Te^s is well-defined for sequencess∈ S_b,e^? . 2. The function Te^s is holomorphic onC−Zfor all sequencess∈ S_b,e^? , it is a uniformly convergent series on every compact subset of C−Z and satisfies, for all s∈ S_b,e^? and all z ∈C−Z:

∂Te^s

∂z (z) = −

l(s)

X

i=1

s_iTe^{s1,···,si−1,si+1,si+1,···,sl(s)}(z) . 3. For all sequences s∈ S_b,e^? and all z∈C−Z we have:

Te^s(−z) = (−1)^||s||Te

←s

(z) .

4. For all z ∈ C−Z , Te^•(z) is symmetrel , that is, for all sequences (ααα;βββ)∈(S_b,e^? )²:

Te^ααα(z)Te^βββ(z) = X

γ

γγ∈she(ααα;βββ)

Te^γγγ(z) .

We will speak respectively of thedifferentiation property and the parity property to refer to the formula of the second point and that of the third point.

19

3. Reduction into monotangent functions

The aim of this section is to show a non trivial link between multitan- gent functions and multizeta values. More precisely, we will show that all (convergent) multitangent functions can be expressed in terms of multizeta values and monotangent functions³. In order to do this, we will proceed classically, that is, we will perform a partial fraction expansion (in the vari- able z) and then sum then after a reorganisation of the terms.

Let us remark that this idea had already been mentioned by Jean Ecalle (cf [14], p. 429) .

3.1. A partial fraction expansion

Let us fix a positive integerr, a family of positive integers s= (si)1≤i≤r

and finaly a family of complex numbers a = (a_i)1≤i≤r, where the a_i are pairwise distinct. Let us also consider the rational fraction defined by

F_a,s(X) = 1

(X+a₁)^s1· · ·(X+a_r)^sr .

We know that the partial fraction expansion of F_a,s(X) can be written in the following way:

F_a,s(X) =

r

X

i=1 si−1

X

j=0

1 j!

F_a^≤i−1_·a^≥i+1_,s^≤i−1_·s^≥i+1(j)

(−a_i) (X+a_i)^si−j .

With the previous notations, an easy computation shows that, for all k∈N, we have:

(−1)^k

k! F_a,s^(k)(X) = X

n1,···,nr≥0 n1+···+nr=k

s1+n1−1 s1−1

· · · ^sr+n_s ^r−1

r−1

(X+a₁)^s1+n1· · ·(X+a_r)^sr+nr . To conclude this subsection, let us introduce three notations:

3Let us recall that a monotangent function is a multitangent function of length 1 .

20

ε^s,k_i = (−1)^{s1+···+si−1+si+1+···+sr+k1+···+ki−1+ki+1+···+kr} ,

iD_k^s(a) =

i−1

Y

l=1

(a_i−a_l)^sl+kl

! _r Y

l=i+1

(a_l−a_i)^sl+kl

! ,

iB_k^s =

i−1

Y

l=1

(−1)^kl

! _r Y

l=i+1

(−1)^sl

!







r

Y

l=1 l6=i

sl+kl−1 s_l−1





.

Of course, in the previous notationsⁱD_k^s(a) andⁱB_k^s, the sequencekhas the same length thana and an i^th index which does not intervene.

So, we finally have the following partial fraction expansion:

F_a,s(X) =

r

X

i=1 si−1

X

k=0

X

k1,···,cki,···,kr≥0 k1+···+cki+···+kr≥0

ε^s,k_i (X+a_i)^si−k

iB_k^s

iD^s_k(a) , (7)

3.2. Expression of a multitangent function in terms of multizeta values and monotangent functions

Plugging (7) in the definition of a multitangent function, we can ex- change the multiple summation (from the definition of a multitangent) with the finite summation (from the partial fraction expansion), because of the absolute convergence, and then sum by decomposing the multiple summa- tion into three terms. Then, the following are successively equal toTe^s(z), if s∈ S_b,e^? :

21

r

X

i=1 si−1

X

k=0

X

k1,···,cki,···,kr≥0 k1+···+cki+···+kr≥0

X

−∞<n_r<···<n₁<+∞

!

ε^s,k_i (z+n_i)^si−k

iB_k^s

iD^s_k(n)

=

r

X

i=1 si−1

X

k=0

X

k1,···,cki,···,kr≥0 k1+···+cki+···+kr≥0

X

ni∈Z

X

(n1;···;ni−1)∈Zi−1 ni<ni−1<···<n1

X

(ni+1;···;nr)∈Nr−i nr <···<ni+1<ni

!

ε^s,k_i (z+n_i)^si−k

iB_k^s

iD_k^s(n)

=

r

X

i=1 si−1

X

k=0

X

k1,···,cki,···,kr≥0 k1+···+cki+···+kr≥0

!









iB^s_kX

ni∈Z

ε^s,k_i (z+n_i)^si−k

X

(ni+1;···;nr)∈Zr−i

−∞≤nr <···<ni+1<ni

1

iD_k^s≥i≥i(n)

X

(n1;···;ni−1)∈Zi−1 ni<ni−1<···<n1≤+∞

1

iD^s≤i

k^≤i(n) !







 .

So, we have the following relation:

Te^s(z) =

r

X

i=1 si−1

X

k=0

Z_i,k^s Te^si−k(z) , where

Z_i,k^s = X

k1,···,cki,···,kr≥0 k1+···+cki+···+kr≥0

iB_k^sZe^{sr+kr,···,si+1+ki+1}Ze^{s1+k1,···,si−1+ki−1} .

The divergent monotangent Te¹ : z 7−→ π

tan(πz) seems to appear in this relation. Nevertheless, the Te¹ coefficient is necessarily null. Indeed, it is not difficult to see that all (convergent) multitangent function decrease exponentially to 0when z −→ +∞ with <e z 6= 0 , for example . So, we obtain:

Theorem 3. (Reduction into monotangent functions, version 1) For all sequence s∈ S_b,e^? , we have:

Te^s(z) =

r

X

i=1 si

X

k=2

Z_i,s^s

i−kTe^k(z) where z ∈C−Z .

22

3.3. Tables of convergent multitangent functions

With a suitable computer algebra software, we can easily generate a table of multitangent functions up to a fixed weight. Different tables can be computed:

1. those given by the previous theorem ;

2. those obtained from the first ones, as soon as we have downloaded a table of exact values of the multizeta values (see [30] for this purpose) ; 3. those obtained from the first ones, as soon as we have downloaded a table of numerical values of the multizeta values (see [2] or [11] for this purpose) ;

4. those obtained from the first ones, after a linearization of products of multizeta values (the choice of linearization by symmetrelity is more natural in this context than using the symmetrality) .

Table 1 contains some examples of such tables. Some boxes in it are empty, which means the expression is the same than in the previous column.

Let us immediately remark that there are a lot ofQ-linear relations between multitangents and all of them all absolutely non trivial. Here are two of them which are easy to state, but the second one is still quite mysterious:

Te^2,1,2 = 0 . (8)

3Te^2,2,2+ 2Te^3,3 = 0 . (9)

We will study this in detail in Section 5.3.

3.4. Linear independence of monotangent functions

At this stage, let us authorize a little incursion in the world of the arithmetic of multitangent functions, a quite obscur world. We have the following lemma. Although it is a simple one, which admits many different proofs, it will be a fundamental lemma which will be used here and there repeatedly in this article.

Lemma 3. The monotangent functions are C-linearly independant.

We give a proof based on the differentiation property of multitangent functions. It is possible to prove this using the Fourier coefficients of mono- tangent functions or by looking at the poles of monotangent functions, etc.

23

Proof. Let us suppose the familly (Teⁿ)n∈N^∗ is not C-free.

So, we would have acces to an integer r ≥ 2, a r-tuple of integers (n₁;· · · ;n_r) satisfying 0 < n₁ < · · · < n_r and a r-tuple of non all zeros complex numbers (λ_n1;· · · ;λ_nr) such that:

r

X

k=1

λ_nkTe^nk = 0 .

Using the differentiation property of multitangent functions, we would obtain:

r

X

k=1

(−1)^nk−1λ_nk (n_k−2)!

∂^nk−1Te¹

∂z^nk−1 = 0 .

So, Te¹ would satisfy a linear differential equation with constant coef- ficients, and therefore could be written as a C-linear combination of expo- nential polynomials. This would allow us to obtain an analytic continuation over allC of the cotangent function.

Because such an analytic continuation is impossible, we have proved that

monotangent functions areC-free.

Since we have just seen that there exists a lot of linear relations be- tween multitangent functions, we know that this lemma can not be extended to multitangent functions. In fact, since we know the Eisenstein relation Te²(z)Te¹(z) = Te³(z), we can affirm that monotangent functions are not algebraically independent, even if we restrict to convergent monotangent functions:

2 Te³2

= 3Te²Te⁴− Te²3

. 3.5. A first approach to algebraic structure of MT GF

Recall that we have denoted byMT GF_CV the algebra, over the field of rational numbers, spanned by all the functionsTe^s ,s∈ S_b,e^? . Now, we will be more precise: for p ∈ N, we will denote by MT GF_CV,p the Q-algebra spanned by all the functions Te^s , with sequences s∈ S_b,e^? of weight p . In the same way, we will denote byMZVCV,p theQ-algebra spanned by all the numbers Ze^s , with sequences s ∈ S_b^? of weight p . Then, MZV_CV will be the Q-algebra spanned by all the numbersZe^s , with sequences s∈ S_b^? .

So, the reduction into monotangent functions, together with the previous lemma, yields the following corollary⁴:

4The notationE·αdenotes the set{e·α;e∈E}.

24